m at h . A P ] 2 3 Ju n 20 09 SELF - SIMILAR BLOW - UP IN PARABOLIC EQUATIONS OF MONGE – AMPÈRE TYPE

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampère equation with power source, with the following basic 2D model (0.1) u t = −|D 2 u| + |u| p−1 u in R 2 × R + , where in two-dimensions |D 2 u| = u xx u yy − (u xy) 2 and p > 1 is a fixed exponent. For a class of " dominated concave " and compactly supported radial initial data u 0 (x) ≥ 0, the Cauchy problem is shown to be locally well-posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of (0.1) in 3D for which |D 2 u| is a cubic operator is u t = |D 2 u| + |u| p−1 u in R 3 × R + , and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similar blow-up in a cubic radial model related to the fourth-order M-A equation of the type u t = −|D 4 u| + u 3 in R 2 × R + , where the cubic operator |D 4 u| is the catalecticant 3 × 3 determinant, is also briefly discussed. This is an earlier extended version of [6], where, in particular, we present a survey on various M-A models; see Appendix A.